Ming Fang 6/12/2009

Outlines  Classical logics  Introduction to DL  Syntax of DL  Semantics of DL  KR in DL  Reasoning in DL  Applications

Classical Logics  Logics are formal languages for representing information such that conclusions can be drawn.

Important Questions  Expressive Power of representation language  able to represent the problem  Soundness of entailment procedure  no false conclusions are drawn  Completeness of entailment procedure  all correct conclusions are drawn  Decidability of entailment problem  there exists a (terminating) algorithm to compute entailment  Complexity  resources needed for computing the solution

Two Familiar Logics  Propositional Logic atomic formula + connectives  propositional formula  First-order Logic atomic formula + connectives + existential and universal quantifiers  well formed formulas

An Example

Introduction to DL  To form a middle ground solution, DL includes some more expressive operations than propositional logic and has decidable or more efficient decision problems than first-order predicate logic  A fragment of FOL  Inherits open-world assumption and non-unique name assumption

Introduction to DL cont’  Originated from frames and semantic networks  Provides formal logical extension  Structured logic

Syntax of DL  Unary predicates: denote concepts e.g. student(Ming)  Binary predicates: denote roles e.g. major(Ming, CS)  FOL constructors: intersection, union, negation, universal quantifier, etc.  Other constructors: inverse, transitivity, etc.  Any (basic) Description Logic is a subset of L3, i.e. the function-free FOL using only at most three variable names

Syntax of DL cont’

Semantics of DL  An atomic concept is interpreted as a set of individuals that is a subset of the domain.  An atomic role is interpreted as a set of pairs of individuals from the domain, i.e., a binary relation over the domain. In this case, if an individual x is related to y via a role R, then y is called an R-successor of x.  The top concept is interpreted as the whole domain.  The bottom concept is interpreted as the empty set.  The interpretation of ¬C is the set of all individuals in the domain which does not belong to the interpretation of C.  Intersection of two concepts C and D is interpreted, as set-intersection i.e., the set of all individuals in the domain that belongs to both the interpretation of C and the interpretation of D.  The value restriction ∀ R.C is interpreted as the set of all individuals in the domain whose R-successors (if any) all belong to the interpretation of C.  The limited existential restriction is interpreted as the set of all individuals in the domain that have at least one R-successor.

KR in DL  A DL KB typically contains two components: TBox and ABox  TBox (terminological box): contains intensional knowledge in the form of a terminology, e.g.  Normally doesn’t change  Assumed to be acyclic

KR in DL cont’  ABox (assertional box): contains extensional knowledge that is specific to individuals, e.g.  Subject to occasional or even constant change  The TBox/ABox distinction is not significant

Reasoning in DL  TBox

Reasoning in DL cont’

 ABox

Applications  OWL  cornerstone of the semantic web for its use in the design of ontologies  OWL DL and Lite are basted on DL  OWL DLP: intersection of DL and Horn Logic Programs. It’s the largest fragment on which the choice for CWA and UNA doesn’t matter

Applications cont’  Configuration  Conceptual Modeling  Query Optimization and View Maintenance  Natural Language Semantics  I3 (Intelligent Integration of Information)  Information Access and Intelligent Interfaces  Terminologies and Ontologies  Software Management  Planning